Find the number of solutions to a pair of equations

key notes:

💡A system of two equations can be classified as follows:

  • If the slopes are the same but the y-intercepts are different, the system has no solution.
  • If the slopes are different, the system has one solution.
  • If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

Three Types of Solutions of a System of Linear Equations:

Consider the pair of linear equations in two variables x and y.

  • a1x + b1y + c1 = 0
  • a2x + b2y + c2 = 0

Here a1, b1, c1, a2, b2, c2 are all real numbers.

Note that, a12 + b12 â‰  0, a22 + b22 â‰  0

1. If (a1/a2) ≠ (b1/b2), then there will be a unique solution. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations.

2. If (a1/a2) = (b1/b2) = (c1/c2), then there will be infinitely many solutions. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables

3. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations.

Learn with an example

How many solutions does the following system have?

  • y = -2x – 4
  • y = 3x + 3

Solution:

Given y = -2x – 4

y = 3x + 3

Rewriting to the general form

-2x – y – 4 = 0

3x – y + 3 = 0

Comparing the coefficients,

(a1/a2) = -â…”

(b1/b2) = -1/-1 = 1

(a1/a2) ≠ (b1/b2)

Hence, this system of equations will have only one solution.

How many solutions do the simultaneous equations below have?

y = 5x + 2

y=7/3x+9/2

  • no solution
  • one solution
  • infinitely many solutions

The first equation is y = 5x + 2, so the slope is 5 and the y-intercept is 2.
The second equation is y =7/3x +9/2, so the slope is 7/3 and the y-intercept is 9/2.

The slopes are different, so the lines intersect at one point. The system has one

How many solutions do the simultaneous equations below have?

y = 6x − 2
y=6x−2/7

  • no solution
  • one solution
  • infinitely many solutions

The first equation is y = 6x − 2, so the slope is 6 and the y-intercept is -2.

The second equation is y = 6x −2/7, so the slope is 6 and the y-intercept is -2/7.

The slopes are the same but the y-intercepts are different, so the lines are
parallel and never intersect. The system has no solution.

Let’s practice: